APPROXIMATION PROPERTIES FOR COSET SPACES AND THEIR
OPERATOR ALGEBRAS
CLAIRE ANANTHARAMAN-DELAROCHE
Abstract. We give a review of the notions of amenability, Haagerup property and
weak amenability for coset spaces Γ/Λ when Λ is an almost normal subgroup of a
discrete group Γ and we discuss the behaviour of the commutant of the corresponding
quasi-regular representation.
Introduction
It is a classical fact that many properties of a discrete group are easily translated into
equivalent properties of its reduced C ∗ -algebra or of its von Neumann algebra. This applies
in particular to the group Γ/Λ whenever Λ is a normal subgroup of a discrete group Γ.
We want to discuss here the case of coset spaces Γ/Λ where Λ is not necessarily normal.
Amenability is the first notion that was studied in this context and is due to Eymard [16].
It extends the notion of amenability of the group Γ/Λ in case Λ is normal but, however,
this notion does not entirely behave as expected. For instance, if Γ1 is a subgroup of
Γ containing Λ, the amenability of Γ/Λ does not always imply the amenability of Γ1 /Λ
[27, 29]. The amenability of Γ/Λ has nothing to do with the injectivity of the von Neumann
algebra λΓ/Λ (Γ)00 , where λΓ/Λ is the quasi-regular representation of Γ on `2 (Γ/Λ) (see
Examples 3.2 and 3.3 below).
When Λ is a normal subgroup, λΓ/Λ (Γ)00 is the left von Neumann algebra L(Γ/Λ) of the
group Γ/Λ and λΓ/Λ (Γ)0 is its right von Neumann algebra R(Γ/Λ), and these two algebras
are isomorphic. This is no longer true in general. As observed by many authors, beginning
with Mackey [26], in the general situation, the algebra λΓ/Λ (Γ)00 , or rather its commutant
λΓ/Λ (Γ)0 , is more directly linked with the coset space CΓ (Λ)/Λ than to Γ/Λ, where CΓ (Λ)
is the commensurator of Λ in Γ (see [13, 3, 2]). We show that λΓ/Λ (Γ)0 is injective when
CΓ (Λ)/Λ is amenable (Proposition 3.5), although the converse is not true (Example 3.3).
The amenability of CΓ (Λ)/Λ is not implied by that of Γ/Λ (Example 3.2). It so appears
that the interesting situation is when Λ is almost normal in Γ, that is, Γ = CΓ (Λ). One
also says that (Γ, Λ) is a Hecke pair. Amenability passes to sub-coset spaces Γ1 /Λ in this
case. This is studied in Section 3.
The analogue for coset spaces of the Haagerup approximation property for groups was
introduced in [4] and also discussed in [30, Remark 3.5]. It makes sense only when Λ is
Key words and phrases. Coset spaces, Hecke algebras, amenability, Haagerup property, weak
amenability.
1
2
CLAIRE ANANTHARAMAN-DELAROCHE
an almost normal subgroup of Γ. It is studied in Section 4 where we show in particular
that it implies that λΓ/Λ (Γ)0 has the Haagerup approximation property with respect to its
canonical state ω. It is important to have in mind that, in contrast with the case where
Λ is a normal subgroup, in general ω is not a trace, and its modular automorphism group
can even exhibit very interesting phase transitions with spontaneous symmetry breaking
as shown in [5].
In Section 5 we introduce and study the notion of weak amenability for Hecke pairs.
This answers a question raised by Quanhua Xu during the 24th International Conference
of Operator Theory in Timisoara (July 2012).
Property (T) for coset spaces is briefly discussed in [30, Remarks 5.6]. Its definition does
not require that Λ is an almost normal subgroup. Although it is not an approximation
property, we discuss this notion in Section 6 because it gives an obstruction for the existence
of almost normal subgroups that yield coset spaces with the Haagerup property.
Finally, the last section is devoted to examples and open problems.
The main tool for the study of notions related to an almost normal subgroup Λ of Γ is
the Schlichting completion (G0 , H 0 ) of (Γ, Λ). It originates in works of G. Schlichting at
the end of the seventies, and was developed by Tzanev [34] (see also [25]). Its advantage is
to reduce the study of the pair (Γ, Λ) to that of (G0 , H 0 ), whose main feature is that H 0 is
a compact open subgroup of G0 . This construction is recalled in Section 2. We observe in
this text that the coset space Γ/Λ has one of the four above mentioned properties if and
only if its associated totally disconnected locally compact group G0 has the corresponding
property.
The first section is not new either, but is developed for the reader’s convenience and to
fix notations. It shows that the commutant λΓ/Λ (Γ)0 of the quasi-regular representation is
generated, as a von Neumann algebra on `2 (Γ/Λ), by the Hecke algebra of the Hecke pair
(CΓ (Λ), Λ), when Λ is any subgroup of Γ. This dates back to the paper [26] of Mackey.
When Λ is almost normal, λΓ/Λ (Γ)0 is called the (right) Hecke von Neumann algebra of
(Γ, Λ), while the norm closure of the Hecke algebra is called its (right) Hecke reduced
C ∗ -algebra.
Although we have added several new contributions, this paper is mainly expository
in nature. The author’s aim is to draw the reader’s attention to these Hecke operator
algebras associated with various kinds of coset spaces (not only the amenable ones), since
they provide lot of interesting examples that deserve to be studied. In particular, by [33,
Théorème 1.25] they provide examples of every type of infinite dimensional von Neumann
factor, including type IIIλ , λ ∈ [0, 1].
Notations and conventions. In this text, Γ will be a discrete group and Λ a subgroup
of Γ. On the other hand, capital Roman letters such as G, H, L will denote general
Hausdorff locally compact groups.
Let H be a closed subgroup of G. Then G/H is the locally compact space of left
cosets and hG/Hi will denote a set of representatives of G/H. Similarly we introduce
APPROXIMATION PROPERTIES FOR COSET SPACES
3
the notation hH \ Gi and hH \ G/Hi for sets of representatives of the spaces H \ G and
H \ G/H of right and double cosets respectively. Elements of G/H will be denoted gH or
ġ. The H-orbit of gH in G/H as well as the corresponding subset of G are written HgH.
Given a right H-invariant function f on G, we shall denote by f˜ the function on G/H
obtained by passing to the quotient. On the other hand, we shall sometimes identify a
function ξ on G/H with the corresponding right H-invariant function on G. This means
in particular that we may write ξ(g) instead of ξ(ġ) for simplicity.
The characteristic function of a subset E of G is as usual denoted by 1E . For g ∈ G,
the function 1gH , when viewed as a function on G/H, will sometimes be written δġ .
1. The commutant of a quasi-regular representation
Recall that the quasi-regular representation λΓ/Λ is defined by
λΓ/Λ (g)ξ(k̇) = ξ(g −1 k̇)
for g, k ∈ Γ and ξ ∈ `2 (Γ/Λ).
Let T ∈ λΓ/Λ (Γ)0 . This operator is completely determined by the function
Tb : g 7→ T (δė )(ġ),
where e is the unit of Γ. Indeed, since T commutes with the quasi-regular representation
we get, for g, k ∈ Γ,
X
T δk̇ =
Tb(k −1 g)δġ .
(1.1)
g∈hΓ/Λi
We immediately see that Tb is Λ-bi-invariant: the right invariance is obvious and the left
invariance follows from the fact that the operator T commutes with the quasi-regular
representation.
2
P
We have g∈hΓ/Λi Tb(g) < +∞, and so Tb(g) = 0 whenever the Λ-orbit of gΛ in Γ/Λ
is infinite. Since
Tc∗ (g) = (T ∗ δė )(ġ) = Tb(g −1 ),
we also see that Tb(g) = 0 whenever the Λ-orbit of g −1 Λ in Γ/Λ is infinite. This implies
that the support of Tb is contained in the commensurator of Λ in Γ, whose definition we
recall now.
1.1. Commensurators and Hecke pairs.
Definition 1.1. Let H be a subgroup of a group G. The commensurator1 of H in G is
the set of g ∈ G such that the subgroup H ∩ gHg −1 has a finite index in H and gHg −1 .
We denote it by CG (H).
1The name of quasi-normalizer is also used.
4
CLAIRE ANANTHARAMAN-DELAROCHE
Since the map h 7→ hgH induces a bijection from H/(H ∩ gHg −1 ) onto the orbit HgH
of gH in G/H, we see H ∩ gHg −1 has a finite index in H if and only if this orbit is finite,
that is, HgH is a finite union of left cosets. It follows that CG (H) is a subgroup of G
which obviously contains the normalizer NG (H) of H in G.
Given g ∈ CH (G), we denote by L(g) the cardinal of the H-orbit of gH and we set
R(g) = L(g −1 ). The function g 7→ L(g)/R(g) is a group homomorphism from CG (H) into
Q∗+ (see [34, Proposition 2.1]).
We say that H is almost normal in G, or that (G, H) is a Hecke pair, if CG (H) = G.
Examples 1.2. Examples of Hecke pairs (G, H) are plentiful. Let us begin by two trivial
ones:
(a) H is a normal subgroup of G. Here L(g) = R(g) = 1 for every g ∈ G;
(b) H is either a finite subgroup or a subgroup of finite index of G.
As we shall see in Section 2, the next example is the prototype of Hecke pairs:
(c) H is a compact open subgroup of a locally compact group G (e.g. G = SLn (Qp )
and H = SLn (Zp ), where Zp is the ring of p-adic integers for a given prime number
p).
Classical examples of Hecke pairs (Γ, Λ) made of countable groups are the following:
(d)
(e)
(f)
(g)
(Γ = SLn (Z[1/p]), Λ = SLn (Z)) ;
(Γ = SLn (Q), Λ = SLn (Z)) ;
(Γ = Q o Q∗+ , Λ = Z o {1}) ;
(Γ = BS(m, n) = ht, x : t−1 xm t = xn i, Λ = hxi).
Here, BS(m; n) is the Baumslag-Solitar group with parameters m, n. Example (f) is the
famous example of Bost and Connes from which they constructed a dynamical system
with spontaneous symmetry breaking.
Remark 1.3. An easy way to construct examples is as follows. Assume that Γ acts by
isometries on a locally finite metric space2 (e.g., by automorphisms of a locally finite
connected graph without loops and multiple edges, when considering the minimal path
length metric). Then, obviously the stabilizer Λ of a vertex is almost normal in Γ. In fact
this is the most general way to construct almost normal subgroups (see [1, Theorem 2.15]).
For instance, the case of the Baumslag-Solitar group, which is an HNN-extension, can be
established by considering its action on the corresponding Bass-Serre tree (see Examples
2.5 for details).
1.2. Hecke algebras and the commutant of the quasi-regular representation.
Let Λ be any subgroup of a group Γ. We denote by C[Γ; Λ] the linear span of the set
{1ΛgΛ : g ∈ CΓ (Λ)}. It is the space of Λ-bi-invariant functions on Γ which are finitely
supported on CΓ (Λ)/Λ (and equivalently on Λ \ CΓ (Λ)/Λ) after passing to the quotient3.
It is easy to guess from the observations preceding Definition 1.1 that this space plays an
2A metric space is locally finite if its balls have a finite number of elements.
3Note that C[Γ; Λ] = C[C (Λ); Λ]. When Λ is almost normal in Γ, the notation H(Γ, Λ) is often used
Γ
instead of C[Γ; Λ].
APPROXIMATION PROPERTIES FOR COSET SPACES
5
important role in the study of λΓ/Λ (Γ)0 . We first recall that it has a natural structure of
involutive algebra, called the Hecke algebra of the Hecke pair (CΓ (Λ), Λ), when equipped
with the convolution product
X
X
(f1 ∗ f2 )(g) =
f1 (g1 )f2 (g1−1 g) =
f1 (gg2−1 )f2 (g2 ),
g1 ∈hΓ/Λi
g2 ∈hΛ\Γi
and the involution
f ∗ (g) = f (g −1 )
(see [22, Section 2.2] for detailed proofs).
Whenever Λ is a normal subgroup of CΓ (Λ), the corresponding Hecke algebra C[Γ; Λ] is
the goup algebra C[CΓ (Λ)/Λ] of the group CΓ (Λ)/Λ and we have
1ΛgΛ ∗ 1ΛhΛ = 1ΛghΛ
for g, h ∈ CΓ (Λ). In general, we get the more complicated formula
X
1
1Λgi hj Λ ,
1ΛgΛ ∗ 1ΛhΛ =
L(gi hj )
gi Λ⊂ΛgΛ, hj ⊂ΛhΛ
where gi Λ ⊂ ΛgΛ means that gi runs over some L(g) representatives of left Λ-cosets
included into ΛgΛ (see [22, page 15] or [34]).
We define a representation R of the opposite algebra of C[Γ; Λ] on `2 (Γ/Λ) by the
formula
X
R(f )ξ(ġ) =
ξ(k̇)f (k −1 g)
k∈hΓ/Λi
2
for f ∈ C[Γ; Λ] and ξ ∈ ` (Γ/Λ).
To check that R(f ) is a bounded operator, it suffices to consider f = 1ΛgΛ with g ∈
CΓ (Λ). We observe that the operator R(f ) is represented, in the canonical basis of `2 (Γ/Λ),
by the matrix [Tḣ,k̇ ] with Tḣ,k̇ = 1ΛgΛ (k −1 h).
P
P
We have suph k∈hΓ/Λi Tḣ,k̇ ≤ R(g) and supk h∈hΓ/Λi Tḣ,k̇ ≤ L(g). It follows that
the matrix [Tḣ,k̇ ] defines a bounded operator from `∞ (Γ/Λ) into itself with [Tḣ,k̇ ]
≤
`∞ →`∞
R(g), and that similarly [Tḣ,k̇ ]
≤ L(g). By the Riesz-Thorin interpolation theorem
1
1
` →`
we get kR(f )k ≤ L(g)1/2 R(g)1/2 .
Straightforward computations show that R(f1 ∗ f2 ) = R(f2 )R(f1 ) and R(f ∗ ) = R(f )∗ .
We are now ready to show the following result ([3, Theorem 2.2]).
Theorem 1.4. The commutant λΓ/Λ (Γ)0 of the quasi-regular representation is the weak
closure N of R(C[Γ; Λ]).
Proof. Obviously, R(C[Γ; Λ]) commutes with λΓ/Λ (Γ). It remains to prove the inclusion
λΓ/Λ (Γ)0 ⊂ N . Let T ∈ λΓ/Λ (Γ)0 . Recall that for y ∈ Γ we have
2
2
X X 2
Tb(g) =
Tb(y −1 g) = kT δẏ k2 < +∞,
g∈hΓ/Λi
g∈hΓ/Λi
6
CLAIRE ANANTHARAMAN-DELAROCHE
and after replacing T by T ∗ ,
2
X Tb(g −1 y) < +∞.
g∈hΓ/Λi
For ξ ∈ `2 (Γ/Λ), we have
Tξ =
X
X
ξ(k̇)
k∈hΓ/Λi
Tb(k −1 g)δġ ,
g∈hΓ/Λi
so
T ξ(ġ) =
X
ξ(k̇)Tb(k −1 g)
k∈hΓ/Λi
where the series is absolutely convergent.
Write CΓ (Λ) as the union of an increasing net Ki where each Ki is a finite union of
double Λ-cosets of elements of CΓ (Λ). Set Ti = 1Ki T ∈ C[Γ; Λ]. If follows from the
above observations that limi k(T − R(Ti ))δẏ k2 = 0 for every y ∈ Γ and that limi T ξ(ġ) −
R(Ti )ξ(ġ) = 0 for every ξ ∈ `2 (Γ/Λ) and g ∈ Γ. Let S ∈ N 0 . Then
lim SR(Ti )δẏ = ST δẏ
i
in k·k2 -norm,
and
lim SR(Ti )δẏ = lim R(Ti )Sδẏ = T Sδẏ
i
Therefore, T ∈ N 00 = N .
i
pointwise.
So, R(C[Γ; Λ]) is weaky dense in the commutant of λΓ/Λ (Γ). The algebra N is called
the (right) von Neumann algebra of the pair (Γ, Λ) and is also written R(Γ, Λ). The norm
closure C∗ρ (Γ, Λ) of R(C[Γ; Λ]) is called the (right) reduced C ∗ -algebra of this pair. When
(Γ, Λ) is a Hecke pair of groups, we shall speak of Hecke von Neumann algebra and Hecke
reduced C ∗ -algebra respectively. When Λ is a normal subgroup of Γ, they are respectively
the right von Neumann algebra R(Γ/Λ) and the right C ∗ -algebra C∗ρ (Γ/Λ) of the group
Γ/Λ.
Corollary 1.5 ([26]). The quasi-regular representation λΓ/Λ is irreducible if and only if
CΓ (Λ) = Λ.
1.3. The canonical state on R(C[Γ; Λ]). We shall denote by ω the vector state on
N = R(C[Γ; Λ]) induced by δė . Note that this state is faithful since δė is separating for
N . However, it is not cyclic when Λ is not a normal subgroup in Γ. We denote by N the
closure of N δė in `2 (Γ/Λ) and by p the orthogonal projection on N . It is an element of
λΓ/Λ (Γ)00 whose central support is one. In particular, the map T 7→ T p is an isomorphism
from N onto the induced von Neumann algebra N p.
It is readily checked that N is the Hilbert subspace of `2(CΓ (Λ)/Λ) consisting in its
Λ-invariant vectors, and that L(g)−1/2 δg̈ : g ∈ hΛ \ CΓ (Λ)/Λi is an orthonormal basis of
N , where δg̈ denote 1ΛgΛ when viewed as a function on Γ/Λ.
APPROXIMATION PROPERTIES FOR COSET SPACES
7
We also observe that T 7→ T δė induces an isometry from L2 (N, ω) onto N : for f1 , f2 ∈
C[Γ; Λ] we have
hR(f1 ), R(f2 )iL2 (N,ω) = hf1 , f2 i`2 (Γ/Λ) .
Thus, N is in standard form on N . We note that, in contrast with the case where Λ is
a normal subgroup of Γ, in general ω is not a trace on N . In [2], Binder proved that ω
is a trace if and only if L(g) = L(g −1 ) for every g ∈ CΓ (Λ). There (and this was further
developed in [5]) the modular automorphism group relative to the state ω on N or C∗ρ (Γ, Λ)
was identified as follows. Set ∆(g) = L(g)/R(g) and view this Λ-bi-invariant function ∆
as the (unbounded) multiplication operator on N , defined by
∆(δg̈ ) = ∆(g)δg̈ ,
∀g ∈ CΓ (Λ).
Then, for T ∈ N = R(Γ, Λ) we have σt (T ) = ∆−it T ∆it (see also [34]). Example 1.2 (f)
has bee studied in detail in [5]. In particular, N is the type III1 hyperfinite factor in this
case.
Remark 1.6. As made obvious above, the commutant of λΓ/Λ (Γ) only depends on the
Hecke pair (CΓ (Λ), Λ). So it will suffice for our purpose to assume in the rest of the paper
that Λ is almost normal in Γ.
2. The Schlichting completion
This completion, which dates back to Schlichting’s papers is described in [34]. It is a
useful tool in order to understand Hecke pairs. For the reader’s convenience, we recall its
construction as well as some of its features that we shall need in the sequel.
Let G be a group and H a subgroup. We say that the pair (G, H) is reduced if the left
action of G on G/H is effective. Replacing G by G1 = G/L and H by H1 = H/L where
L = ∩g∈G gHg −1 , we get an effective action of the group G1 on the coset space G1 /H1 .
This latter space is in canonical bijective correspondance with G/H. Clearly, H is almost
normal in G if and only if H1 is almost normal in G1 . There is no loss of generality to
assume effectiveness.
Proposition 2.1. Let Λ be an almost normal subgroup of Γ. There exists a unique (up
to isomorphism) triple (G0 , H 0 , θ) such that the pair (G0 , H 0 ) is reduced and
(a) G0 is a locally compact group, H 0 is a compact open subgroup of G0 ;
(b) θ is a homomorphism from Γ into G0 with θ(Γ) = G0 and θ−1 (H 0 ) = Λ.
Proof. We may assume that the pair (Γ, Λ) is reduced, without loss of generality. Then Γ
is viewed as a subgroup of the group Bij(Γ/Λ) of bijections of Γ/Λ. Let us denote by G0
the closure of Γ in the space of all maps from Γ/Λ into itself, equipped with the topology
of pointwise convergence.
We first check that G0 is contained into Bij(Γ/Λ). Obviously, the elements of G0 are
injective maps, since Γ/Λ is discrete. We now take f ∈ G0 and show that f is surjective.
Let (gi ) be a net in Γ such that f = limi gi in the topology of pointwise convergence. Let
i0 be such that f (ė) = gi ė for i ≥ i0 . We set g = gi0 . So hi = g −1 gi ∈ Λ whenever i ≥ i0 .
Observe that limi hi = g −1 f . In particular, f1 = g −1 f is an injective map and for y ∈ Γ/Λ,
8
CLAIRE ANANTHARAMAN-DELAROCHE
h ∈ Λ, we have f1 (hy) = (g −1 f )(hy) = limi hi hy ⊂ Λy. Hence, f1 (Λy) ⊂ Λy. Since Λy is
a finite set, it follows that y is in the range of f1 . Thus, we see that f1 ∈ Bij(Γ/Λ), and
f ∈ Bij(Γ/Λ) too.
Now, we want to show that G0 is a locally compact group. For x ∈ Γ/Λ, we set
= {g 0 ∈ G0 : g 0 (x) = x}. It is easily checked that G0ė is the closure of Λ. Let us prove
that Λ is relatively compact. This follows from the fact that Λ is a subset of Πy∈Γ/Λ Λy,
which is compact, by the Tychonov theorem. Since finite intersections of such stabilizers
G0x form a basis of neighbourhoods of the identity in G0 , we see that G0 is a locally compact
group. Note that the G0x are both compact and open, and so G0 is totally disconnected.
G0x
We set H 0 = G0ė and denote by θ the inclusion of Γ into G0 . We immediately see that
(G , H 0 , θ) fulfils the required conditions. Note that H 0 = Λ.
0
Let (G01 , H10 , θ1 ) be another triple with the same properties. Observe that G01 = θ1 (Γ)H10
and G0 = θ(Γ)Λ. Moreover, the map α : Γ/Λ → G01 /H10 sending ġ onto θ1 (g)H10 ∈ G01 /H10
is a bijection, which induces an effective action ι of G01 on Γ/Λ. We have ι(θ1 (g)) = θ(g)
for g ∈ Γ and so ι(G01 ) ⊂ G0 . Moreover, ι(H10 ) is contained into the stabilizer of ė, that
is Λ. On the other hand, we have Λ ⊂ ι(H10 ), and therefore Λ = ι(H10 ). It follows that ι
is an isomorphism from G01 onto G0 . Moreover it is an homeomorphism because it sends
continuously the compact open set H10 onto the open set Λ.
The pair (G0 , H 0 ) is called the Schlichting completion of (Γ, Λ). We keep this notation
in the rest of the paper. We shall use the following obvious facts.
Lemma 2.2. The Schlichting completion (G0 , H 0 ) has the following properties:
(i) θ(Λ) = H 0 ;
(ii) G0 = θ(Γ)H 0 and for every g 0 ∈ G0 there exists a unique left Λ-coset gΛ such that
g 0 = θ(g)h0 with h0 ∈ H 0 .
e 1 Λ) =
(iii) θ induces a bijection θe : gΓ → θ(g)H 0 from Γ/Λ onto G0 /H 0 such that θ(gg
e 1 Λ) for g, g1 ∈ Γ, that is, a Γ-equivariant bijection. This induces a unitary
θ(g)θ(g
operator U : `2 (Γ/Λ) → `2 (G0 /H 0 ) such that U λΓ/Λ (g) = λG0 /H 0 (θ(g))U .
(iv) ψ 0 7→ ψ = ψ 0 ◦ θ is a bijection from the space of (continuous) right H 0 -invariant
functions on G0 onto the space of right Λ-invariant functions on Γ. The inverse
bijection is ψ 7→ ψ 0 where ψ 0 (g 0 ) = ψ(g) for g 0 = θ(g)h0 .
Similarly, for double cosets we shall use the following lemma.
Lemma 2.3. The map ΛgΛ 7→ H 0 θ(g)H 0 is a bijection from the set Λ \ Γ/Λ onto H 0 \
G0 /H 0 . It follows that ψ 0 7→ ψ = ψ 0 ◦ θ is a bijection from the space of (continuous) H 0 bi-invariant functions on G0 onto the space of Λ-bi-invariant functions on Γ. Moreover,
ψ 0 is positive definite on G0 if and only if ψ is positive definite on Γ.
Proof. Obviously the map ΛgΛ 7→ H 0 θ(g)H 0 is surjective. Let us show that it is injective.
Assume that H 0 θ(g)H 0 = H 0 θ(k)H 0 . Then, we have θ(g)H 0 θ(k)−1 ∩ H 0 6= ∅. Since
θ(g)H 0 θ(k)−1 is open, it contains some θ(h) with h ∈ Λ and so θ(g)H 0 = θ(hk)H 0 . By
condition (b) of Proposition 2.1, we see that k −1 h−1 g ∈ Λ and therefore ΛgΛ = ΛkΛ.
APPROXIMATION PROPERTIES FOR COSET SPACES
9
Assume now that ψ is positive definite. We take λ1 , . . . , λn in C and g10 , . . . , gn0 ∈ G0
and we write gi0 = θ(gi )h0i . Then we have
X
X
λi λj ψ 0 ((gi0 )−1 gj0 ) =
λi λj ψ(gi−1 gj ) ≥ 0.
1≤i,j≤n
1≤i,j≤n
The converse is obvious.
Remark 2.4. One may defines the Hecke algebra of the pair (G0 , H 0 ) exactly as for (Γ, Λ).
The bijection of the previous lemma induces an isomorphism of involutive algebras from
C[G0 ; H 0 ] onto C[Γ; Λ].
Examples 2.5. If Λ is a normal subgroup of Γ, its Schlichting completion is (Γ/Λ, {ė}).
The Schlichting completion of (SL(n, Z[1/p]), SLn (Z)) is (SLn (Qp ), SLn (Zp )).
The Schlichting completion for Examples 1.2 (e), (f) involves more sophisticated objects
from number theory, namely the ring of finite adèles and its subring of integers (see [5, 33]).
Example 1.2 (g) is a particular case of HNN-extension. Consider a group Λ and an
isomorphism σ : H → K between two subgroups of Λ. Denote by Γ the HNN-extension
relative to (Λ, H, σ), that is, Γ has the following presentation
Γ = Λ, t|t−1 ht = σ(h), ∀h ∈ H .
(The group BS(m, n) corresponds to Λ = Z, H = mZ, K = nZ and σ(mz) = nz for every
z ∈ Z.)
The group Γ acts transitively, with the obvious action, on its Bass-Serre tree T , which
is defined as follows (see [31]): the set T 0 of vertices is Γ/Λ and the set T 1 of edges
is (Γ/H) t (Γ/K), the sources of gH and gK are gΛ, their targets are gtΛ and gt−1 Λ
respectively. In particular each vertex has degree [Λ : H] + [Λ : K]. So, T is locally finite
if and only if the subgroups H and K have a finite index in Λ. In this case, as observed
in Remark 1.3, (Γ, Λ) is a Hecke pair, since Λ is a vertex stabilizer. Then, the Schlichting
completion of (Γ, Λ) is described as a particular case of the following result.
Proposition 2.6. Let Γ be a group acting by isometries on a locally finite metric space X
and Λx be the stabilizer of an element x ∈ X. We endow the group Iso(X) of isometries
of X with the topology of pointwise convergence and denote by θ : Γ → Iso(X) the group
homomorphism corresponding to the action. The Schlichting completion of (Γ, Λx ) is the
pair (G/L, H/L) where G and H are the closures of θ(Γ) and θ(Λx ) respectively in Iso(X)
and where L = ∩g∈G gHg −1 .
Proof. Since X is locally finite, it is known that Iso(X) is a locally compact group which
acts properly on X (see [1] for instance). Therefore the group G defined above acts properly
on X, and H, which is easily seen to be the stabilizer of x, is compact. It is open since X
is discrete. The pair (G0 = G/L, H 0 = H/L) is reduced and the conditions of Proposition
2.1 are obviously satisfied.
Remark 2.7. We observe that the group G in the above proposition does not depend on
the choice of x ∈ X and therefore the group G0 of the Schlichting completion of (Γ, Λx )
only depends on x through the quotient of G by a compact normal subgroup.
10
CLAIRE ANANTHARAMAN-DELAROCHE
3. Co-amenability
Co-amenability is the first relative property that was studied for a pair (G, H) where
H is a closed subgroup of the locally compact group G. It is due to Eymard [16] and can
be defined by several properties he proved to be equivalent. We just recall the following
ones.
Theorem 3.1 ([16]). The following properties are equivalent:
(i) there exists a G-invariant mean on L∞ (G/H), that is, a G-invariant state on
L∞ (G/H) ;
(ii) the trivial representation of G is weakly contained into λG/H , that is, for every
2
ε > 0 and every compact
subset K
of G, there exists a unit function ξ in L (G/H)
such that sups∈K λG/H (s)ξ − ξ ≤ ε ;
2
(iii) the pair (G, H) has the conditional fixed point property, that is, for any compact
convex subset Q of a locally convex topological vector space, if G acts continuously
and affinely on Q in such a way that there is a H-fixed point, then there also exists
a G-fixed point.
When these properties are satisfied, we say that H is co-amenable in G, or that the
coset space G/H is amenable. Other terminologies found in the literature are that the
pair (G, H) is amenable or that H is co-Følner in G. In case H is reduced to the identity,
we say that G is amenable.
When H is a normal subgroup of G, the amenability of the coset space G/H is the
same as the amenability of the group G/H. In the discrete case and when Λ is a normal
subgroup of Γ, it is well-known that the group Γ/Λ is amenable if and only if the reduced
group C ∗ -algebra of Γ/Λ is nuclear, or equivalently, if and only if its von Neuman algebra
is injective. Without the assumption of normality, these algebras are respectively replaced
by C∗ρ (Γ, Λ) and R(Γ, Λ). We now give examples showing that the amenability of Γ/Λ
cannot in general be read on these algebras.
Example 3.2. In [27] (see also [29]), we find the following nice example of co-amenable
subgroup. Consider any non-trivial discrete group Q. Set Λ = ⊕n≥0 Q, Γ1 = ⊕n∈Z Q,
Γ = Γ1 o Z = Q o Z, the wreath product of Q by Z. Monod and Popa proved that Λ
is co-amenable in Γ, whatever Q is. As a consequence, one sees that co-amenability is
not hereditary in the following sense: Γ/Λ is amenable but Γ1 /Λ is amenable only if Q is
amenable. This fact contrasts with the hereditary property that holds when Λ is a normal
subgroup of Γ. Here the commensurator of Λ in Γ is Γ1 and Γ1 /Λ is the group ⊕n<0 Q. It
follows that λΓ/Λ (Γ)0 , which is isomorphic to the right von Neumann algebra of the group
Γ1 /Λ, is isomorphic to the von Neumann tensor product ⊗n<0 R(Q), where R(Q) is the
right von Neumann algebra of the group Q. It is injective if and only if this group Q is
amenable.
Example 3.3. Consider the Hecke pair (Γ = SL2 (Q), Λ = SL2 (Z)). Its Hecke algebra is
abelian (see [5]), hence the von Neumann algebra λΓ/Λ (Γ)00 is of type I. However, Λ is not
co-amenable in Γ. Otherwise its Schlichting completion G0 would be an amenable locally
compact group (see below). But G0 = SL2 (A), where A is the ring of finite adèles, is not
amenable.
APPROXIMATION PROPERTIES FOR COSET SPACES
11
Another example of the same kind is the Hecke pair (Γ = SLn (Z[1/p]), Λ = SLn (Z)).
Its Schlichting completion is (G0 = SLn (Qp ), H 0 = SLn (Zp )), which is a Gelfand pair.
It follows that its Hecke pair is abelian (indeed an algebra of polynomials by the Satake
isomorphism [22, page 19]), although G0 is not amenable.
Proposition 3.4. Let (Γ, Λ) be a Hecke pair and (G0 , H 0 ) its Schlichting completion. The
following conditions are equivalent:
(i) Λ is co-amenable in Γ;
(ii) H 0 is co-amenable in G0 ;
(iii) G0 is amenable.
Proof. This result is contained in [34, Proposition 5.1]. Let us first recall the proof of (i)
⇒ (ii). One uses the characterization of co-amenability in terms of conditional fixed point
property. Let G0 y C be a continuous affine action on a compact convex set which has a
H 0 -fixed point. Since Λ is co-amenable in Γ we deduce the existence of a θ(Γ)-fixed point
in C, and since θ(Γ) is dense in G0 , this point is also G0 -fixed. To prove the converse,
we observe that if the trivial representation of G0 is weakly contained in the quasi-regular
representation λG0 /H 0 , then it follows immediately from Lemma 2.2 (iii) that the trivial
representation of Γ is weakly contained in λΓ/Λ .
The equivalence between (ii) and (iii) is obvious, using the fixed point characterizations
and the fact that H 0 is compact.
Proposition 3.5. Let Λ be a subgroup of Γ which is co-amenable in its commensurator.
Then C∗ρ (Γ, Λ) is nuclear and R(Γ, Λ) is injective, and thus λΓ/Λ (Γ)00 is injective.
As already observed, it suffices to consider the case where Λ is almost normal in Γ. One
way to prove this result is to observe that C∗ρ (Γ, Λ) and R(Γ, Λ) are corners of C∗ρ (G0 ) and
R(G0 ) respectively, where C∗ρ (G0 ) is the (right) reduced C ∗ -algebra of G0 and R(G0 ) its
weak closure (see [34]). We give here another proof. For further purposes, we state first
an easy fact, more general that what is immediately needed.
Lemma 3.6. Let Λ be an almost normal subgroup of Γ and let ψ be a Λ-bi-invariant
function from Γ to C. We assume that there exist two bounded Dfunctions ξ,
E η from Γ/Λ
into a Hilbert space K such that for g, h ∈ Γ we have ψ(h−1 g) = ξ(ġ), η(ḣ) . Then there
exists a unique normal map Ψ : R(Γ, Λ) → R(Γ, Λ) such that Ψ(R(f )) = R(ψf ) for every
f ∈ C[Γ; Λ]. Moreover Ψ is completely bounded4 with kΨkcb ≤ kξk∞ kηk∞ and whenever
2
ξ = η, then Ψ is completey positive with kΨkcb = kξk∞ .
Proof. We define two bounded operators S, T : `2 (Γ/Λ) → `2 (Γ/Λ, K) by
Sl(ġ) = ξ(ġ)l(ġ),
T l(ġ) = η(ġ)l(ġ).
4We denote by k·k
cb its completely bounded norm.
12
CLAIRE ANANTHARAMAN-DELAROCHE
Then, straightforward computations show that, for f ∈ C[Γ; Λ], we have
E
X D
ξ(ġ), η(ḣ) l(ḣ)f (h−1 g))
S ∗ R(f ) ⊗ 1 T l(ġ) =
h∈hΓ/Λi
X
=
l(ḣ)ψ(h−1 g)f (h−1 g))
h∈hΓ/Λi
= R(ψf )l(ġ).
The other assertions follow immediately.
Note that the restriction of Ψ to C∗ρ (Γ, Λ) is a completely bounded map from this
C -algebra into itself.
∗
We observe that ω ◦ Ψ = ψ(e) ω and
ω
Ψ(R(f )) − R(f )
∗
2
Ψ(R(f )) − R(f ) = kψf − f k`2 (Γ/Λ) .
(3.2)
We set kψkcb = inf kξk∞ kηk∞ where (ξ, η) runs over all pairs satisfying the properties
of the lemma. We have kΨkcb ≤ kψkcb . When Λ is a normal subgroup, it is well-known
that these two quantities are the same (see [23] for instance). We do not know whether
this is true in general.
Our second observation is a characterization of co-amenability in terms of positive
definite functions when the subgroup is almost normal. No such characterization exists
for general subgroups. We begin by giving a sufficient condition for a positive definite
function to be a coefficient of a quasi-regular representation. This extends a result of
Godement [18] stating that a square integrable continuous positive definite function on a
locally compact group is a coefficient of its regular representation.
Theorem 3.7. Let H be a closed subgroup of a locally compact group G such that there
exists a G-invariant measure on G/H. Let ϕ be a continuous positive definite function on
G. We assume that ϕ is H-bi-invariant and that ϕ̃ obtained by passing to the quotient is
in L2 (G/H). Then ϕ is a coefficient of the quasi-regular representation.
Proof. We have fixed a left-invariant measure µ on G/H, denoted also d ṫ. We use arguments similar to those of Godement’s proof, which concerns the case where H is the
trivial subgroup {e}. Given a continuous function with compact support ξ ∈ Cc (G/H),
we define the function ρ(ϕ)ξ by
Z
ρ(ϕ)ξ(ṡ) =
ϕ(s−1 t)ξ(ṫ) dṫ.
G/H
This function belongs to L2 (G/H). Indeed, if K is a compact subset of G/H containing
the support of ξ, by using the Cauchy-Schwarz inequality, the invariance of µ and the fact
APPROXIMATION PROPERTIES FOR COSET SPACES
that ϕ(s) = ϕ(s−1 ), we have
Z
Z
2
|ρ(ϕ)ξ(ṡ)| dṡ ≤
G/H
Z
G/H
≤
13
2
ϕ(s−1 t)ξ(ṫ) dṫ dṡ
G/H
Z
G/H×G/H
2
2
2
2
1K (ṫ)ϕ(s−1 t) dṫ dṡ kξk2 = µ(K)kϕk2 kξk2 .
Therefore, ρ(ϕ) is an operator which has the space Cc (G/H) of compactly supported
continuous functions on G/H as domain. It is unbounded in general and non-negative,
since for every ξ ∈ Cc (G/H) we have, due to the fact that ϕ is positive definite,
Z
ξ(ṡ)ϕ(s−1 t)ξ(ṫ) dṫ dṡ ≥ 0.
hξ, ρ(ϕ)ξi =
G/H×G/H
We still denote by ρ(ϕ) its Friedrichs extension. It is a non-negative self-adjoint operator.
If ξ ∈ Cc (G/H) and s ∈ G, a routine computation shows that
λG/H (s)ρ(ϕ)ξ = ρ(ϕ)λG/H (s)ξ
and therefore λG/H (s) commutes with the Friedrichs extension.
Let (Vi ) be a decreasing net of compact neighbourhoods of ė in G/H,
such that ∩i Vi =
R
{ė}. We set ξi = ρ(ϕ)1/2 fi where fi is a non-negative function with G/H fi (ṫ) d ṫ = 1 and
such that fi (ṫ) = 0 outside Vi . We have
Z
hξi , ξj i = hfi , ρ(ϕ)fj i =
fi (ṫ)ϕ(t−1 s)fj (ṡ) dṫ dṡ,
G/H×G/H
and so
Z
fi (ṫ)ϕ(t−1 s) − ϕ(e)fj (ṡ) dṫ dṡ.
|hξi , ξj i − ϕ(e)| ≤
G/H×G/H
We remark that ϕ, being continuous and positive definite is uniformly continuous. It
follows that limi,j hξi , , ξj i = ϕ(e), from which we deduce that (ξi ) is a Cauchy net. We
denote by ξ its limit.
We shall now show that ϕ is the coefficient of λG/H associated with ξ. We have
ξ, λG/H (s)ξ = lim ξi , λG/H (s)ξi
i
D
E
= lim ρ(ϕ)1/2 fi , λG/H (s)ρ(ϕ)1/2 fi
i
= lim fi , λG/H (s)ρ(ϕ)fi
i
Z
= lim
fi (ṫ)ϕ(t−1 su)fi (u̇) dṫ du̇,
i
since λG/H (s) commutes with ρ(ϕ).
G/H×G/H
14
CLAIRE ANANTHARAMAN-DELAROCHE
We have
Z
−1
fi (ṫ)ϕ(t su)fi (u̇) dṫ du̇ − ϕ(s)
G/H×G/H
Z
≤
fi (ṫ)fi (u̇)ϕ(t−1 su) − ϕ(s) dṫ du̇.
G/H×G/H
Using again the uniform continuity of ϕ, we see that
ξ, λG/H (s)ξ = lim ξi , λG/H (s)ξi = ϕ(s).
i
Note that is ϕ(s) = ξ, λG/H (s)ξ for all s, then ϕ is H-bi-invariant if and only if the
vector ξ is H-invariant. If moreover ϕ̃ is compactly supported, then there exists a compact
subset K of G such that the support of ϕ is contained in KH.
We are now able to give a characterization of co-amenability in terms of positive definite
functions when Λ is almost normal in Γ.
Theorem 3.8. Let Λ be an almost normal subgroup of Γ. The following conditions are
equivalent:
(i) Λ is co-amenable in Γ ;
(ii) for every ε > 0 and every finite subset K of Γ, there exists a Λ-invariant
unit
function ξ with finite support on Γ/Λ such that supg∈K λΓ/Λ (g)ξ − ξ 2 ≤ ε ;
(iii) for every ε > 0 and every finite subset K of Γ, there exists a Λ-bi-invariant positive
definite function on Γ whose support is a finite union of double cosets, such that
supg∈K |ϕ(g) − 1| ≤ ε.
Proof. (ii)
that (ii) holds. Given ε > 0 and K, let ξ be as in (ii). We set
⇒ (iii). Assume
ϕ(g) = ξ, λΓ/Λ (g)ξ . Then ϕ is Λ-bi-invariant and we have
sup |ϕ(g) − 1| ≤ sup λΓ/Λ (g)ξ − ξ ≤ ε.
g∈K
2
g∈K
(iii) ⇒ (i) is also very easy, thanks to Theorem 3.7 and the classical inequality
λΓ/Λ (g)ξ − ξ 2 ≤ 2 ξ, λΓ/Λ (g)ξ − ξ .
2
It remains to prove (i) ⇒ (ii). We shall use the Schlichting completion and the following
easy lemma.
Lemma 3.9. Let H be a compact and co-amenable subgroup of a locally compact group
G. For every ε > 0 and every compact subset
K of G, there exists a H-invariant unit
function ξ in L2 (G/H) such that supg∈K λG/H (g)ξ − ξ 2 ≤ ε. Moreover we may choose
ξ to be compactly supported.
Proof. We may assume that K contains H. Take ε0 < 1/2 such that 5ε0 ≤ ε. There exists
a unit vector η ∈ L2 (G/H) such that
sup λG/H (g)η − η ≤ ε0 .
g∈K
2
APPROXIMATION PROPERTIES FOR COSET SPACES
15
R
We set η 0 = H λG/H (h)η dh, the integration being with respect to the Haar probability
measure on H. We have kη 0 − ηk2 ≤ ε0 and kξ − ηk2 ≤ 2ε0 , where ξ = η 0 /kη 0 k2 . Then ξ is
H-invariant and supg∈K λG/H (g)ξ − ξ 2 ≤ 5ε0 ≤ ε. Moreover, if we have started with a
compactly supported function η, so is ξ.
Proof of (i) ⇒ (ii) in Theorem 3.8. By Proposition 3.4, H 0 is co-amenable in G0 . Then by
0
2
0
0
Lemma 3.9, there
is a net (ξi ) of H
-invariant unit vectors in ` (G /H ), finitely0 supported,
such that limi λG0 /H 0 (s)ξi − ξi 2 = 0 uniformly on compact subsets of G . Let U :
`2 (Γ/Λ) → `2 (G0 /H 0 ) be as in Lemma 2.2 (iii). Then the net (U −1 ξi ) is made
of Λ
invariant finitely supported unit vectors such that limi λΓ/Λ (s)U −1 ξi − U −1 ξi 2 = 0
pointwise.
Theorem 3.8 has the following immediate consequence.
Corollary 3.10. Let Λ be an almost normal co-amenable subgroup of Γ Then Λ is coamenable in any subgroup Γ1 of Γ containing Λ.
This is obvious with the characterization (iii) of Theorem 3.8. This result has to be
compared with Example 3.2. Proposition 3.5 is also an easy consequence of this characterization.
Proof of Proposition 3.5. Let (ϕi ) be a net of Λ-bi-invariant positive definite functions
on Γ, supported on finite unions of double cosets, that converges to 1 pointwise.
We
may assume that ϕi (e) = 1. Each ϕi can be written as ϕi (g) = ξi , λΓ/Λ (g)ξi where
ξi is a unit Λ-invariant vector of `2 (Γ/Λ). By Lemma 3.6 there exists a normal unital
completely positive map Φi from R(Γ, Λ) into itself such that Φi (R(f )) = R(ϕi f ) for every
f ∈ C[Γ, Λ]. Obviously, Φi has a finite rank. It remains to show that for T ∈ R(Γ, Λ) we
have limi Φi (T ) = T in the weak topology. By approximation it suffices to take T = R(f ).
The conclusion follows from the equality (see (3.2)),
X
2
2
2
kΦi (R(f )) − R(f )kL2 =
|ϕi (g) − 1| |f (g)| .
g∈hΓ/Λi
4. Co-Haagerup property
The now called Haagerup property was first detected by this author for free groups [21].
Since then, this property proved to be very useful in various branches of mathematics (see
[8]). Close connections with approximation properties of finite von Neumann algebras
were discovered in [10, 9]). We begin by recalling some definitions.
Definition 4.1. We say that a locally compact group G has the Haagerup (approximation)
property if there exists a net (ϕi ) of normalized5 continuous positive definite functions on
G, vanishing at infinity, which converges to 1 uniformly on compact subsets of G.
5i.e. such that ϕ (e) = 1.
i
16
CLAIRE ANANTHARAMAN-DELAROCHE
Let M be a von Neumann algebra, equipped with a normal faithful state ω. A completely positive map Φ : M → M such that ω ◦ Φ = ω gives rise to a bounded operator
b : L2 (M, ω) → L2 (M, ω) in the following way. Given x ∈ M , we denote by x
Φ
b this element
when we want to stress the fact that x is viewed as an element of L2 (M, ω). We have
ω(Φ(x)∗ Φ(x)) ≤ kΦk ω(Φ(x∗ x)) ≤ kΦk ω(x∗ x).
[
b is well defined by Φ(b
b x) = Φ(x).
Then Φ
Definition 4.2. We say that M has the Haagerup (approximation) property with respect
to ω if there exists a net (Φi ) of unital completely positive maps from M to M such that
(a) ω ◦ Φi = ω for every i ;
ci is a compact operator for every i ;
(b) Φ
(c) limi kΦi (x) − xkL2 (M,ω) = 0 for every x ∈ M .
This property has been previously defined under the assumption that ω is a trace. In
this case, Jolissaint proved that it does not depend on the choice of the faithful tracial
state [24]. Moreover, when M is the tracial von Neumann algebra L(Γ) of a group Γ, then
Γ has the Haagerup property if and only if L(Γ) has the Haagerup property [9].
Given an almost normal subgroup Λ of Γ, the notion of Haagerup property relative to Λ
appears in [30, Remarks 3.5] (see also [12]). No definition is known without the assumption
of almost normality. Observe that this latter property is implied by Conditions (a) and
(b) of the next definition.
Definition 4.3. Let Λ be an almost normal subgroup of a group Γ. We say that Λ is
co-Haagerup in Γ, or that the coset space Γ/Λ, or the Hecke pair (Γ, Λ), has the Haagerup
property, if there exists a net (ϕi ) of normalized positive definite functions on Γ such that
(a) for all i the function ϕi is Λ-bi-invariant and, passing to the quotient, ϕ̃i is in the
space c0 (Γ/Λ) of functions on Γ/Λ that vanish at infinity;
(b) limi ϕi = 1 pointwise.
Proposition 4.4. Let Λ be an almost normal subgroup of Γ which is co-Haagerup. Then
R(Γ, Λ) has the Haagerup property with respect to its canonical state ω.
Proof. Let (ϕi ) be a net of positive definite functions as in Definition 4.3. By the classical
GNS construction, for each i there is a representation (πi , Ki ) of Γ and a unit vector
ξi ∈ Ki such that ϕi (g) = hξi , πi (g)ξi i for g ∈ Γ. Morever, ξi is Λ-invariant since ϕi is Λ-biinvariant. By Lemma 3.6, there is a unique normal completely positive map Φi from N =
R(Γ, Λ) into itself such that Φi (R(f )) = R(ϕi f ) for f ∈ C[Γ; Λ]. Moreover Φi preserves
the state ω, and as in the proof of Proposition 3.5 we see that limi kΦi (T ) − T kL2 = 0 for
every T ∈ R(Γ, Λ).
Recall that L(g)−1/2 δg̈ : g ∈ hΛ \ Γ/Λi is an orthonormal basis of L2 (N, ω). We have
ci (δg̈ ) = ϕi (g)δg̈ . It follows that Φ
ci is a compact operator since ϕi , when viewed as a
Φ
function on Λ \ Γ/Λ, belongs to c0 (Λ \ Γ/Λ).
Proposition 4.5. Let (Γ, Λ) be a Hecke pair and (G0 , H 0 ) its Schlichting completion. Then
Γ/Λ has the Haagerup property if and only if the group G0 has the Haagerup property.
APPROXIMATION PROPERTIES FOR COSET SPACES
17
Proof. Assume that Γ/Λ has the Haagerup property. Let (ϕi ) be a net of positive definite
functions on Γ as in Definition 4.3. We define ϕ0i on G0 by the expression ϕ0i (g 0 ) = ϕi (g),
where gΛ is the unique element of Γ/Λ such that g 0 H 0 = θ(g)H 0 . It is a H 0 -bi-invariant
positive definite function on G0 , by Lemma 2.3. Since H 0 is compact, we see that ϕ0i
vanishes at infinity. Obviously, the net (ϕ0i ) converges to 1 uniformly on compact subsets
of G0 , and therefore this group has the Haagerup property.
Conversely, assume that G0 has the Haagerup property. Then there exists a unitary C0 representation6 π on a Hilbert space H, which contains weakly the trivial representation
(see [8, Theorem 2.1.2]). Let 0 < ε ≤ 1/2 and a compact subset K of G0 containing H 0
be given.
There exists a unit vector ξ ∈ H such that supg∈K kπ(g)ξ − ξk ≤ ε. We set
R
ξ 0 = H 0 π(h)ξ dh (where we integrate with respect to the Haar probability measure on
H 0 ) and ξ 00 = ξ 0 /kξ 0 k. Then we have kξ 0 − ξk ≤ ε and kξ 00 − ξk ≤ 2ε. It follows that
the coefficient ϕ0 of π defined on G0 by ϕ0 (g) = hξ 00 , π(g)ξ 00 i is H 0 -bi-invariant, vanishes at
infinity and satisfies, for g ∈ K,
|ϕ0 (g) − 1| ≤ kπ(g)ξ 00 − ξ 00 k ≤ 5ε.
By passing to the quotient, we get ϕe0 ∈ c0 (G0 /H 0 ).
We define a Λ-bi-invariant positive definite function ϕ on Γ by
ϕ(g) = ϕ0 (θ(g)).
Since ϕ
e is obtained from ϕe0 via the natural bijection from Γ/Λ onto G0 /H 0 , we see that
ϕ
e ∈ c0 (Γ/Λ).
Now, starting with a net (ξi ) of almost invariant unit vectors in H (i.e. such that
limi kπ(g)ξi − ξi k = 0 uniformly on compact subsets of G0 ), this construction provides
us with a net (ϕi ) of Λ-bi-invariant positive definite functions vanishing at infinity on Γ,
which converges to one pointwise.
5. Weak co-amenability
The notion of weak amenability stems also from the seminal paper [21] although the
terminology was introduced later [14]. Let G be a locally compact group and A(G) its
Fourier algebra. Recall that A(G) is the predual of the von Neumann algebra L(G) of
G. A multiplier ψ of A(G) is said to be a completely bounded multiplier if its transposed
operator is completely bounded. We denote by kψkcb the completely bounded norm of this
operator. A completely bounded multiplier ψ is characterized by the following property:
there exist two continuous bounded functions ξ, η from G into some Hilbert space K such
that ψ(k −1 g) = hξ(g), η(k)i for g, k ∈ G. Moreover we have kψkcb = inf kξk∞ kηk∞ where
ξ, η runs over all possible such decompositions (see [6], and [23] for a simple proof).
Definition 5.1. A locally compact group G is said to be weakly amenable if there exists
a net (ψi ) in A(G) which converges to 1 uniformly on compact subsets of G and such
that there exists a constant C with kψi kcb ≤ C for all i. The Cowling-Haagerup constant
Λcb (G) of G is the infimum of such C for all possible such nets.
6Recall that a representation is C if all its coefficients vanish at infinity.
0
18
CLAIRE ANANTHARAMAN-DELAROCHE
In fact, by [14, Proposition 1.1], we may assume that the ψi ’s have a compact support.
The previous definition has to be compared with Leptin’s characterization of amenability, which requires the stronger condition that supi kψi kA(G) < +∞. Recall that kψi kcb ≤
kψi kA(G) .
Definition 5.2. We say that a C ∗ -algebra (resp. a von Neumann algebra) B has the
completely bounded approximation property (CBAP) (resp. the weak* completely bounded
approximation property (W*CBAP)) if there exists a net of finite rank maps Ψi : B → B
(resp. normal finite rank maps) that converges to the identity map in the point-norm
(resp. point-weak*) topology and such that supi kΨi kcb ≤ C.
The Haagerup constant Λcb (B) is the infimum such C for all possible such nets.
Let Γ be a discrete group. Then Γ is weakly amenable if and only if its reduced C ∗ algebra has the CBAP, and also if and only if its von Neumann algebra has the W*CBAP.
Moreover, the corresponding constants are the same. For a proof of this result due to
Haagerup, see [7, Theorem 12.3.10].
Definition 5.3. Let Λ be an almost normal subgroup of a group Γ. We say that Λ is
weakly co-amenable in Γ, or that the coset space Γ/Λ, or the Hecke pair (Γ, Λ), is weakly
amenable, if there exists a net (ψi ) of complex-valued functions on Γ such that
(a) for all i the function ψi is Λ-bi-invariant and, by passing to the quotient, ψ̃i has a
finite support on Γ/Λ ;
(b) for all i there exist two bounded
functions
ξi , ηi from Γ/Λ into some Hilbert space
D
E
Ki such that ψi (k −1 g) = ξi (ġ), ηi (k̇) for all g, k ∈ Γ ;
(c) limi ψi = 1 pointwise;
(d) supi kψi kcb ≤ C.
We denote by Λcb (Γ, Λ) the infimum such C for all possible such nets7 .
Proposition 5.4. Let Λ be a weakly co-amenable almost normal subgroup of Γ. Then
R(Γ, Λ) has the W*CBAP and C∗ρ (Γ, Λ) has the CBAP . Moreover we have Λcb (R(Γ, Λ)) ≤
Λcb (Γ, Λ) and Λcb (C∗ρ (Γ, Λ)) ≤ Λcb (Γ, Λ).
Proof. Let (ψi ) be a net satisfying the conditions of Definition 5.3 for the pair (Γ, Λ). Let
Ψi be the completely bounded function from R(Γ, Λ) into itself constructed from ψi in
Lemma 3.6. This map has a finite rank and kΨi kcb ≤ kψi kcb . Moreover, again as in the
proof of Proposition 3.5, we see that limi kΨi (T ) − T kL2 = 0 for every T ∈ R(Γ, Λ).
Proposition 5.5. Let (Γ, Λ) be a Hecke pair and (G0 , H 0 ) its Schlichting completion.Then
Γ/Λ is weakly amenable if and only if the group G0 is weakly amenable. In this case, we
have Λcb (G0 ) = Λcb (Γ, Λ).
Proof. Let ψ, ξ, η as in Lemma 3.6 and let ψ 0 be the corresponding H 0 -bi-invariant function
on G0 . Recall that ψ 0 (g 0 ) = ψ(g) where g 0 = θ(g)h0 , h0 ∈ H 0 . We set
ξ 0 (θ(g)H 0 ) = ξ(gΛ),
η 0 (θ(g)H 0 ) = η(gΛ).
7Recall that kψ k = inf kξ k kη k
i cb
i ∞
i ∞ where (ξi , ηi ) runs over all pairs as in (b).
APPROXIMATION PROPERTIES FOR COSET SPACES
19
For g 0 = θ(g)h0 and g10 = θ(g1 )h01 , we have
ψ 0 ((g10 )−1 g 0 ) = ψ(g1−1 g) = hξ(gΛ), η(g1 Λ)i
= hξ 0 (g 0 H 0 ), η 0 (g10 H 0 )i = hξ 00 (g 0 ), η 00 (g10 )i,
where ξ 00 (g 0 ) = ξ 0 (g 0 H 0 ) and similarly for η 00 .
These functions ξ 00 and η 00 being continuous and bounded on G0 by kξk∞ and kηk∞
respectively, we conclude that ψ 0 is a completely bounded multiplier of G0 with kψ 0 kcb ≤
kψkcb .
e
Since, after passing to quotient, ψe0 (θ(g)H 0 ) = ψ(gΛ))
for g ∈ Γ, we easily conclude that
0
e
ψ is compactly supported whenever ψ has a finite support.
Finally, if (ψi ) is a net as in Definition 5.3, we see that the corresponding net (ψi0 ) converges to 1 uniformly on compact subsets of G0 , and it follows from the above observations
that G0 is weakly amenable. Moreover, we get Λcb (G0 ) ≤ Λcb (Γ, Λ).
To prove the converse, let us first start with a completely bounded multiplier ψ 0 of G0
such that for some compact subset K of G0 and some ε > 0 we have |ψ 0 (h0 kh00 ) − 1| ≤ ε
for k ∈ K, h0 , h00 ∈ H 0 . We denote by ξ 0 , η 0 two continuous bounded functions from G0
into some Hilbert space H such that ψ 0 (k −1 g) = hξ 0 (g), η 0 (k)i for g, k ∈ G0 .
R
We set ψ 00 (g) = H 0 ×H 0 ψ 0 (h0 gh00 ) dh0 dh00 where the integrations are with respect to
the Haar probability measure on H 0 . Of course, ψ 00 is H 0 -bi-invariant, and we have
supk∈K |ψ 0 (k) − ψ 00 (k)| ≤ ε. If ξ 00 and η 00 are defined on G0 /H 0 by
Z
Z
ξ 00 (gH 0 ) =
ξ 0 (gh0 ) dh0 , η 00 (gH 0 ) =
η 0 (gh0 ) dh0 ,
H0
00
we have ψ (k
−1
00
0
H0
00
0
0
g) = hξ (gH ), η (kH )i for g, k ∈ G .
Now, if we define ψ : Γ → C and ξ, η : Γ/Λ → H by
ψ(g) = ψ 00 (θ(g)),
ξ(gΛ) = ξ 00 (θ(g)H 0 ),
η(gΛ) = η 00 (θ(g)H 0 ),
we see that ψ is a Λ-bi-invariant completely bounded multiplier on Γ such that ψ(k −1 g) =
hξ(gΛ), η(kΛ)i for g, k ∈ Γ. Moreover, if ψ 0 is compactly supported, so is ψ 00 and the
support of ψe is finite.
Finally, starting from a net (ψi0 ) of compactly supported completely bounded multipliers
of G0 with supi kψi0 kcb = c < +∞, converging to 1 uniformly on compact subsets of G0 , we
get from the above considerations a net (ψi ) which satisfies the conditions of Definition
5.3 with supi kψi kcb ≤ c. Hence we have Λcb (Γ, Λ) ≤ Λcb (G0 ).
6. Co-rigidity
We first recall the now classical notion of property (T) for a locally compact group G.
Let (π, H) be a unitary representation of a locally compact group G. Given ε > 0 and
a compact subset K of G, recall that an (ε, K)-invariant vector is a unit vector ξ ∈ H
such that supg∈K kπ(g)ξ − ξk ≤ ε. One says that π almost contains invariant vectors (or
that the trivial representation of G is weakly contained in π) if it contains (ε, K)-invariant
vectors for every ε > 0 and every compact subset K of G.
20
CLAIRE ANANTHARAMAN-DELAROCHE
Definition 6.1. We say that G has the Kazdhan property (T), or is rigid, if every unitary
representation of G that almost contains invariant vectors actually contains a non-zero
invariant vector.
Proposition 6.2. The following conditions are equivalent:
(i) G is rigid;
(ii) for every 0 < δ < 2 there exist ε > 0 and a compact subset K of G such that for
every unitary representation (π, H) of G and every unit (ε, K)-invariant vector
ξ ∈ H, there is a G-invariant unit vector η with kξ − ηk ≤ δ ;
(iii) every net (ϕi ) of normalized continuous positive definite functions on G that converges to 1 uniformly on compact subsets of G also converges to 1 uniformly on
G.
For proofs, we refer for instance to [15, Proposition 1.16 and Théorème 5.11].
Definition 6.3. Let H be a closed subgroup of a locally compact group G. We say that
a unitary representation of G almost contains invariant vectors which are H-invariant if
for every ε > 0 and every compact subset K of G, there is a H-invariant vector which is
(ε, K)-invariant.
We say that G has property (T) relative to H (or that H is co-rigid in G, or that the
coset space G/H has the property (T)) if every unitary representation of G which almost
contains invariant vectors which are H-invariant actually contains a non-zero G-invariant
vector.
Proposition 6.4. The following conditions are equivalent:
(i) H is co-rigid in G;
(ii) for every 0 < δ < 2 there exist ε > 0 and a compact subset K of G such that
for every unitary representation (π, H) of G and every unit (ε, K)-invariant and
H-invariant vector ξ ∈ H, there is a G-invariant unit vector η with kξ − ηk ≤ δ.
(iii) every net (ϕi ) of normalized H-bi-invariant continuous positive definite functions
on G that converges to 1 uniformly on compact subsets of G also converges to 1
uniformly on G.
The proof is similar to that of Proposition 6.2.
Proposition 6.5. Let Λ be an almost normal subgroup of Γ. Then the coset space Γ/Λ
has the property (T) if and only if the group G0 has the property (T).
Proof. Assume that G0 has Property (T). Let (ϕi ) be a net of Λ-bi-invariant positive
definite functions on Γ that converges pointwise to 1. The corresponding net (ϕ0i ) on G0
(see Lemma 2.3) converges to 1 uniformly on compact subsets of G0 and so uniformly on
G0 , from which is follows that (ϕi ) converges to 1 uniformly on Γ.
Conversely, assume that the coset space Γ/Λ has the property (T). Let (π, H) be a
unitary representation of G0 that almost contains invariant vectors. Using the fact that
H 0 is compact, it is easy to see that it almost contains invariant vectors which are H 0 invariant. Then the representation π ◦ θ of Γ almost contains invariant vectors which
APPROXIMATION PROPERTIES FOR COSET SPACES
21
are Λ-invariant. It follows that π has a non-zero θ(Γ)-invariant vector, which is also G0 invariant by the density of θ(Γ) in G0 .
Remark 6.6. Reference to this notion of co-rigidity is made in [30, Remarks 5.6]. It should
not be confused with the more familiar notion of rigid inclusion of a closed subgroup H in a
locally compact group G, due to Kazhdan and Margulis, which reads as follows: H ⊂ G is
rigid if every unitary representation of G which almost contains invariant vectors actually
contains a non-zero H-invariant vector.
7. Examples and problems
7.1. Amenable coset spaces. Note that a coset space Γ/Λ is automatically amenable
when Γ is amenable. On the other hand, if Γ has the property (T), then Γ/Λ is amenable
if and only if Λ has a finite index in Γ.
It was left as an open problem by Greenleaf whether the existence of a reduced pair8
(Γ, Λ) such that Γ/Λ is amenable implies the amenability of Γ [19, Problem, page 18].
The first examples were found only some 20 years later by van Douwen: he proved that
every finitely generated non-abelian free group contains such a co-amenable subgroup.
Later, other examples were provided in [27, 29] (see 3.2). Thereafter, this problem has
been studied by many researchers [17, 20, 28]. Most of their examples are built on free or
amalgamed free products of groups.
As for reduced Hecke pairs (Γ, Λ) such that Γ/Λ is amenable, there is the famous
example (Γ = Q o Q∗+ , Λ = Z o {1}) studied by Bost and Connes. But it is amenable for
the obvious reason that Γ is amenable.
Problem 1. Greenleaf’s problem is still open in this setting : find examples of reduced
Hecke pairs (Γ, Λ) such that Γ/Λ is amenable without Γ being so.
7.2. Coset spaces with the Haagerup property. We first observe that the Haagerup
property of Γ/Λ has nothing to do with the fact that Γ possesses or not this property. For
instance any finitely generated group is a quotient of a free group, and free groups have
the Haagerup property. On the other hand, SL2 (Z), which has the Haagerup property is
a quotient of Z2 o SL2 (Z) which has not this property.
An easy way of constructing examples of Hecke pairs (Γ, Λ) such that Γ/Λ has the
Haagerup property is as follows.
Proposition 7.1. Assume that Γ acts on a locally finite tree T and let Λ be the stabilizer
of some vertex v0 . Then Λ is co-Haagerup in Γ.
Proof. Denote by d the length metric on T . The function ψ : g 7→ d(v0 , gv0 ) is conditionally
negative definite and Λ-bi-invariant. Moreover it is a proper function after passing to the
quotient on Γ/Λ. Then the sequence (ϕn ), where ϕn = exp(−ψ/n) satisfies the properties
of Definition 4.3.
8In order to avoid trivial examples we are only interested in reduced pairs. Indeed, otherwise it is easy to
construct artificially many examples by starting with a surjective homomorphism α from a non-amenable
group Γ onto an amenable group Γ0 , then take any subgroup Λ0 of Γ0 and consider (Γ, Λ = α−1 (Λ0 )).
22
CLAIRE ANANTHARAMAN-DELAROCHE
For instance, let Γ be the HNN-extension relative to (Λ, H, σ) where σ : H → K is an
isomorphism between two subgroups of finite index in Λ (see Examples 2.5). Then Γ/Λ
has the Haagerup property.
Another example is given by (SL2 (Z[1/p]), SL2 (Z)). One may use the action of SL2 (Z[1/p])
on the (p + 1)-regular tree. Another proof is to observe that the group G0 of the Schlichting completion of this pair is SL2 (Qp ) which has the Haagerup property and then use
Proposition 4.5.
7.3. Weakly amenable coset spaces. The same preliminary observation applies: the
weak amenability of Γ/Λ has nothing to do with that of Γ.
Proposition 7.2. Assume that Γ acts on a locally finite tree T and let Λ be the stabilizer
of some vertex v0 . Then Λ is weakly co-amenable in Γ and Λcb (Γ, Λ) = 1.
Proof. Using Proposition 5.5, it suffices to show that the group G0 of the Schlichting
completion of (Γ, Λ) is weakly amenable with constant Λcb (Γ) = 1. By Proposition 2.6,
we have G0 = G/L where G acts properly on the tree T and L is a normal compact
subgroup of G. We know that G and G/L are simultaneously weakly amenable with the
same constant Λcb (see [14, Proposition 1.3]). To conclude, we use Theorem 6 of [32] which
states that a locally compact group acting on tree is weakly amenable with constant 1 as
soon as the stabilizer of one vertex is compact.
Thus the pair (SL2 (Z[1/p]), SL2 (Z)) and the HNN-extensions of the previous subsection
provide weakly amenable coset spaces with constant 1.
Problem 2. Let Λ be a weakly co-amenable subgroup of Γ. We have shown in Proposition
5.4 that Λcb (R(Γ, Λ)) ≤ Λcb (Γ, Λ). Are there examples where the inequality is strict ?
Same question for Λcb (C∗ρ (Γ, Λ)). Have we Λcb (C∗ρ (Γ, Λ)) = Λcb (R(Γ, Λ))?
7.4. Co-rigidity. Obviously, if Γ has the property (T), then every subgroup Λ is co-rigid
in Γ. In contrast with the amenable case, it is easy to construct co-rigid reduced examples
where Γ has not the property (T) and may even be amenable. The following example
is due to S. Popa [30, Remark 5.6.2’]. It is based on a simple observation: assume that
there exist a finite subset F of Γ and an integer n ≥ 1 such that every g ∈ Γ belongs to
some g1 Λg2 Λ · · · gn Λ with gi ∈ F for i = 1, . . . , n. Then Λ is co-rigid in Γ. Indeed, let
(π, H) be a unitary representation of Γ such that there exists a unit Λ-invariant vector
ξ ∈ H with supg∈F kπ(g)ξ − ξk ≤ 1/2n. Then we have kπ(g)ξ − ξk ≤ 1/2 for every g ∈ Γ,
and the element of smallest norm in the closed convex envelope of π(Γ)ξ is non-zero and
Γ-invariant. An example of this kind is Γ = Q o Q∗+ , Λ = Q∗+ and F = (1, 1), since
ΛF Λ = Γ.
In the setting of Hecke pairs, the situation is quite different. Assume that for such a pair
(Γ, Λ), the coset space Γ/Λ has the property (T). Whenever Γ is amenable or even only has
the Haagerup property, it follows from Theorem 3.8 (or Definition 4.3) and Proposition
6.4 that Λ has a finite index in Γ.
Problem 3. Find examples of reduced Hecke pairs (Γ, Λ) such that Γ/Λ has the property
(T) without Γ having this property.
APPROXIMATION PROPERTIES FOR COSET SPACES
23
When Λ is a normal subgroup of Γ, then Γ/Λ has the property (T) if and only if its
von Neumann algebra has this property [11].
Problem 4. When Λ is almost normal and co-rigid in Γ, is there an analogue of property
(T) for R(Γ, Λ) ?
7.5. Final remark. Let Γ be acting by isometries on a locally finite metric space X and
denote by Λx the stabilizer of x ∈ X. We have observed in Remark 2.7 that the group G0
of the Schlichting completion of (Γ, Λx ) is the quotient of a group G, independent of x by a
normal compact subgroup which may depend on x. The four properties considered in this
paper (amenability, Haagerup property, weak amenability, rigidity) are stable by passing
to such quotients. Since the corresponding properties of Γ/Λx are read on G0 (Propositions
3.4, 4.5, 5.5, 6.5) we conclude that they do not depend on the choice of x. More precisely,
with the notation of Proposition 2.6, we see that Γ/Λx has one of the above mentioned
properties if and only if the closure G of θ(Γ) in Iso(X) has the same property.
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Laboratoire de Mathématiques-Analyse, Probabilités, Modélisation-Orléans (MAPMO - UMR6628),
Fédération Denis Poisson (FDP - FR2964), CNRS/Université d’Orléans, B. P. 6759, F-45067
Orléans Cedex 2
E-mail address: claire.anantharaman@univ-orleans.fr